Pre Calculus Variation Direct and Inverse

In this lesson, we are going to talk about variation - both direct variation and inverse variation. Variation is a way of talking about how different things relate to each other. Depending on the type of variation, we'll know what form the relationship takes. However, we should note that variation isn't connected very closely to functions. While you can describe it in the language of functions, it's easier to talk about variation with equations, so that's what we'll do. You'll learn how two things can be directly related to each other, or they can be inversely related to each other. You'll also learn about the joint variation.

Variation Direct and Inverse

Variation is a way of talking about how different things relate to each other. Depending on the type of variation, we'll know what form the relationship takes. However, we should note that variation isn't connected very closely to functions.
While you can describe it in the language of functions, it's easier to talk about variation with equations, so that's what we'll do.

Direct variation says that two things are directly related to each other. If one goes up, the other goes up. If one goes down, the other goes down. They (usually) go at different rates, but the same "direction".

There are many different ways to say two things (let's call them x and y) are in direct variation:

x and y vary directly;

y varies directly as x;

x and y are directly proportional;

y is directly proportional to x.

In any case, all these phrases mean the same thing mathematically:

y = k ·x,

where k is a constant. It is called the proportionality constant or the constant of variation. It is the rate at which the two things are connected.

Inverse variation is the opposite of direct variation. It says that two things are inversely related to each other. If one goes up, the other goes down. If one goes down, the other goes up. They (usually) go at different rates, but opposite "directions".
[Caution: inverse variation is not related to inverse functions. It's based on the idea of multiplicative inverses, like 3 and [1/3].]

Like direct variation, there are many alternative ways to say that x and y are in inverse variation:

x and y vary inversely;

y varies inversely as x;

x and y are inversely proportional;

y is inversely proportional to x.

Not only that, but the relationship is sometimes called a reciprocal proportion or (confusingly) indirect variation. Still, they all mean the same thing mathematically:

y =

k

x

,

where k is a constant. It serves the same purpose as it does for direct variation: it gives the rate at which the things are connected.

Joint variation is a variation where multiple direct variations are happening at the same time. We could say:

z varies jointly as x and y;

z is jointly proportional to x and y.

Both of these would mean

z = k·x ·y,

where k is once again a proportionality constant.

We can also combine direct and inverse variation if we have these sorts of relationships going on simultaneously. Pay attention to what kinds of variation each variable gives, then put them all together. Notice that no matter how many variations are
put together, you only need a single constant k.

Variation Direct and Inverse

x and y vary directly. When x = 15, y=3. What is x when y=12?

Direct variation implies a relationship of the form y = k·x, where k is some constant.

Plugging into y=k·x, we can solve for k:

3 = k ·15

Once we know that k=[1/5], we have y = [1/5]·x. Now we can find out what x is when y=12.

x=60

m and n are directly proportional. When m = 14, n=10. What is n when m=20?

Directly proportional means the same thing as direct variation. Direct variation implies a relationship of the form y = k·x, where k is some constant. In this case, we're using different variables, but the format is still the same.

Plugging into n=k·m, we can solve for k:

10 = k ·14

Once we know that k=[5/7], we have n = [5/7]·m. Now we can find out what n is when m=20.

n=[100/7]

x and y vary inversely. If x=6 when y = 5, what is y when x=3?

Inverse variation implies a relationship of the form y = [k/x], where k is some constant.

Plugging into y = [k/x], we can solve for k:

5 =

k

6

Once we know that k=30, we have y = [30/x]. Now we can find out what y is when x=3.

y=10

a and b are inversely proportional. If a=9 when b = 11, what is a when b=30?

Inversely proportional means the same thing as inverse variation. Inverse variation implies a relationship of the form y = [k/x], where k is some constant. In this case, we're using different variables, but the format is still the same.

Plugging into b = [k/a], we can solve for k:

11 =

k

9

Once we know that k=99, we have b = [99/a]. Now we can find out what a is when b=30.

a=[33/10]

z varies jointly as x and y. When x = 3 and y=9, then z=135. What is y when x=8 and z=200?

Joint variation implies a relationship of the form z = k·x·y, where k is some constant.

Plugging into z = k·x·y, we can solve for k:

135 = k ·3 ·9

Once we know that k=5, we have z = 5 ·x ·y. Now we can find out what y is when x=8 and z=200.

y=5

a varies directly as b and inversely as c. When b=15 and c=30, then a=25. What is b when a=6 and c=25?

In this problem, we are combining multiple types of variation. a varies directly as b translates to a = k ·b, while a varies inversely as c translates to a = [k/c].
Combining these two as the problem does, we have a relationship of the form

a = k ·

b

c

,

where k is a constant. [Notice that no matter how many variations are combined, we only ever have a single proportionality constant k.]

Plugging into a = k ·[b/c], we can solve for k:

25 = k ·

15

30

Once we know that k=50, we have a = 50 ·[b/c]. Now we can find out what b is when a=6 and c=25.

b=3

The acceleration (a) of an object is directly proportional to the force acting on the object (F). If an object has an acceleration of 2 [(m/s)/s] when it has a force of 18 N acting on it, what force will give it an acceleration of 10 [(m/s)/s]?

Directly proportional means the same thing as direct variation. Direct variation implies a relationship of the form y = k·x, where k is some constant. In this case, we're using different variables, but the format is still the same.

From the problem statement, we have the equation F = k·a. We have that a=2 and F=18 makes one pair, so we can plug those in to find k:

18 = k·2

Once we know that k=2, we have F = 9a. From there, we can plug in a=10 to find out what F needs to be:

F = 9·10

[Remember to give F the appropriate units from the problem statement.]

A force of 90 N.

For a rectangle of fixed area, the length is inversely proportional to the width. If the length of the rectangle is 10 cm when the width is 15 cm, what must the width be when the length is 25 cm?

Begin by naming the variables: l = length and w=width.

Inversely proportional means the same thing as inverse variation. Inverse variation implies a relationship of the form y = [k/x], where k is some constant. In this case, we're using different variables, but the format is still the same.

We are working with the equation w = [k/(l)]. We have that l = 10 and w=15 makes one pair, so we can plug those in to find k:

15 =

k

10

Once we know that k=150, we have w = [150/(l)]. From there, we can plug in l = 25 to find out what w must be:

w =

150

25

[Remember to give w the appropriate units from the problem statement.]

The width must be 6 cm when the length is 25 cm.

The force of gravity (F) between two objects is jointly proportional to the masses of each object (m1 and m2) while also inversely proportional to the square of the distance between the two objects (r). Express this relationship as a mathematical equation.

We need to turn the problem statement into an equation. We have two kinds of variation in the problem statement: joint variation (z = k·y·x) and inverse variation (y = [k/x]).

We are also combining the two types of variation. Both will occur, but remember there is only ever one proportionality constant k.

Notice that the problem says "inversely proportional to the square of the distance between the two objects (r)." This means we won't use just r, but instead its square: r2.

F = k·[(m1·m2)/(r2)]

The kinetic energy of an object is jointly proportional to the mass of the object and the square of the object's velocity. If an object begins with some mass and velocity, then the velocity doubles while the energy remains constant, what fraction of the original mass must the object have?

Begin by creating variables for the things we are working with: E=kinetic energy, m=mass, v=velocity.

Once you have variables, a general formula can be created to express the relationship. It is a joint variation and it uses the square of the velocity, so we have

E = k·m ·v2

Notice that the problem never gave us any numbers to work with. We don't have specific things to substitute into the formula. That's okay. We can still name those things that we would have plugged in.
We can break the problem down into two moments: before the increase in velocity, and after the increase in velocity. The energy remains constant, so we can use E for both. The velocity doubles, so we can use v for the original velocity, and 2v for the later velocity. From the way the question is phrased, we know the mass will change, but we don't know by how much. Use m to denote the original mass and mnew to denote the new mass.

We can express each moment as an equation using our formula:

Before: E = k ·m ·v2After: E = k ·mnew ·(2v)2

We want to know what mnew is. Because we have E on the left side of each equation, we can set the right sides equal to each other:

k ·m ·v2 = k ·mnew ·(2v)2

Simplify the equation, cancel out those variables you can, then solve for mnew in terms of m (the original mass).

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

Variation Direct and Inverse

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